dc.contributor.advisor Richard P. Stanley. en_US dc.contributor.author Chebikin, Denis en_US dc.contributor.other Massachusetts Institute of Technology. Dept. of Mathematics. en_US dc.date.accessioned 2008-12-11T18:28:30Z dc.date.available 2008-12-11T18:28:30Z dc.date.copyright 2008 en_US dc.date.issued 2008 en_US dc.identifier.uri http://hdl.handle.net/1721.1/43793 dc.description Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. en_US dc.description Includes bibliographical references (p. 75-76). en_US dc.description.abstract We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation [sigma] = [sigma] 1 [sigma] 2 an defined as the set of indices i such that either i is odd and ai > ui+l, or i is even and au < au+l. We show that this statistic is equidistributed with the 3-descent set statistic on permutations [sigma] = [sigma] 1 [sigma] 2 ... [sigma] n+1 with al = 1, defined to be the set of indices i such that the triple [sigma] i [sigma] i + [sigma] i +2 forms an odd permutation of size 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials ... using alternating descents. By looking at the number of alternating inversions in alternating (down-up) permutations, we obtain a new qanalog of the Euler number En and show how it emerges in a q-analog of an identity expressing E, as a weighted sum of Dyck paths. Other parts of this thesis are devoted to polytopes relevant to the descent statistic. One such polytope is a "signed" version of the Pitman-Stanley parking function polytope, which can be viewed as a generalization of the chain polytope of the zigzag poset. We also discuss the family of descent polytopes, also known as order polytopes of ribbon posets, giving ways to compute their f-vectors and looking further into their combinatorial structure. en_US dc.description.statementofresponsibility by Denis Chebikin. en_US dc.format.extent 76 p. en_US dc.language.iso eng en_US dc.publisher Massachusetts Institute of Technology en_US dc.rights M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. en_US dc.rights.uri http://dspace.mit.edu/handle/1721.1/7582 en_US dc.subject Mathematics. en_US dc.title Polytopes, generating functions, and new statistics related to descents and inversions in permutations en_US dc.type Thesis en_US dc.description.degree Ph.D. en_US dc.contributor.department Massachusetts Institute of Technology. Dept. of Mathematics. en_US dc.identifier.oclc 261341960 en_US
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